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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 160808, 15 pages
http://dx.doi.org/10.1155/2012/160808
Research Article

Derivatives of the Berezin Transform

LATP, UMR CNRS 6632, CMI, UniversitΓ© de Provence, 39 rue Fjoliot-Curie, 13453 Marseille Cedex 13, France

Received 3 July 2009; Accepted 21 December 2011

Academic Editor: MiroslavΒ Englis

Copyright © 2012 Hélène Bommier-Hato. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For a rotation invariant domain Ξ©, we consider 𝐴2(Ξ©,πœ‡) the Bergman space and we investigate some properties of the rank one projection 𝐴(𝑧)∢=βŸ¨β‹…,π‘˜π‘§βŸ©π‘˜π‘§. We prove that the trace of all the strong derivatives of A(z) is zero. We also focus on the generalized Fock space 𝐴2(πœ‡π‘š), where πœ‡π‘š is the measure with weight π‘’βˆ’|𝑧|π‘š, π‘š>0, with respect to the Lebesgue measure on ℂ𝑛 and establish estimations of derivatives of the Berezin transform of a bounded operator T on 𝐴2(πœ‡π‘š).

1. Introduction and Statement of the Main Results

We consider a rotation invariant open set Ξ© in ℂ𝑛 and πœ‡ a positive rotation invariant measure on Ξ©; we suppose that πœ‡ has moments of every order. Let 𝐿2(Ξ©,πœ‡) be the Hilbert space of square integrable complex-valued functions on Ξ© and π’œ2(Ξ©,πœ‡) its subspace consisting of holomorphic elements. We assume that for each compact set πΎβŠ‚Ξ© there exists 𝐢=𝐢(𝐾) such that for all π‘“βˆˆπ’œ2(Ξ©,πœ‡)supπ‘§βˆˆπΎ||||𝑓(𝑧)≀𝐢‖𝑓‖𝐿2(Ξ©,πœ‡).(1.1) It is known that π’œ2(Ξ©,πœ‡) is a closed space of 𝐿2(Ξ©,πœ‡) and possesses a reproducing kernel 𝐾(πœ”,𝑧)∢=𝐾𝑧(πœ”): we haveξ€œπ‘“(𝑧)=Ω𝑓(πœ”)𝐾(𝑧,πœ”)π‘‘πœ‡(πœ”)=βŸ¨π‘“,πΎπ‘§βŸ©,(1.2) for all π‘“βˆˆπ’œ2(Ξ©,πœ‡) and π‘§βˆˆΞ©.

For a bounded linear operator 𝑇 on π’œ2(Ξ©,πœ‡), the Berezin transform of 𝑇 is the function 𝑇 defined on Ξ© by𝑇(𝑧)∢=βŸ¨π‘‡π‘˜π‘§,π‘˜π‘§βŸ©,(1.3) where π‘˜π‘§ is the normalized reproducing kernelπ‘˜π‘§(πœ”)∢=𝐾(πœ”,𝑧)𝐾(𝑧,𝑧)1/2.(1.4) The case of the Fock space, where Ξ©=ℂ𝑛 and πœ‡2 is the Gaussian measure, was considered by Coburn, Englis, and Zhang. Coburn [1] has shown that 𝑇 is a Lipschitz function. Namely, for π‘₯,π‘¦βˆˆβ„‚π‘›,||𝑇𝑇||||||,(π‘₯)βˆ’(𝑦)≀2‖𝑇‖π‘₯βˆ’π‘¦(1.5) the constant 2 being sharp (see [2]).

Englis and Zhang [3] have shown that 𝑇 has bounded derivatives of all orders. Namely, for any multi-indices 𝛽, 𝛾, there exists a constant 𝑐𝛽,𝛾, depending on 𝛽, 𝛾, and 𝑛 only, such thatβ€–β€–πœ•π›½πœ•π›Ύξ‚π‘‡β€–β€–βˆžβ‰€π‘π›½,𝛾‖𝑇‖,(1.6) where πœ•π›½ will stand forπœ•|𝛽|πœ•π‘§π›½11β‹―πœ•π‘§π›½π‘›π‘›πœ•,whereπœ•π‘§π‘—=12ξ‚΅πœ•πœ•π‘₯π‘—πœ•βˆ’π‘–πœ•π‘¦π‘—ξ‚Άfor𝑧𝑗=π‘₯𝑗+𝑖𝑦𝑗(1.7) and similarly for πœ•. Recently, the author extended Coburn’s result to weighted Fock spaces, corresponding to Ξ©=ℂ𝑛 endowed with the measureπ‘‘πœ‡π‘š(𝑧)=π‘’βˆ’|𝑧|π‘šπ‘‘π‘‰(𝑧),(1.8) where π‘š is a positive parameter and 𝑑𝑉 is the normalized Lebesgue measure on ℂ𝑛, such that the volume of the unit ball in ℂ𝑛 is equal to 2𝑛. We have showed that 𝑇 satisfies a local Lipschitz condition. There exist positive constants 𝐴, 𝐢, depending on 𝑛 and π‘š only, such that, for any π‘Žβˆˆβ„‚π‘› with |π‘Ž|>𝐴, there is a neighbourhood π‘ˆ of π‘Ž that satisfies||𝑇𝑇||(π‘Ž)βˆ’(π‘₯)≀𝐢‖𝑇‖|π‘Ž|(π‘š/2)βˆ’1|π‘₯βˆ’π‘Ž|,βˆ€π‘₯inπ‘ˆ.(1.9) In this paper we will investigate some properties of the derivatives of the Berezin transform on π’œ2(Ξ©,πœ‡). For 𝑧 in Ξ©, if 𝐴(𝑧) is the rank one projection𝐴(𝑧)∢=βŸ¨β‹…,π‘˜π‘§βŸ©π‘˜π‘§,(1.10) we have𝑇(𝑧)=tr(𝑇𝐴(𝑧)).(1.11) We first fix some notations. Let ℕ𝑛0 denote the set of all 𝑛-tuples with components in the set β„•0 of all nonnegative integers. If 𝛽=(𝛽1,…,𝛽𝑛)βˆˆβ„•π‘›0, we let |𝛽|∢=𝛽1+β‹―+𝛽𝑛 denote the length of 𝛽 and 𝛽! stands for βˆπ‘›π‘—=1𝛽𝑗!. If 𝛾=(𝛾1,…,𝛾𝑛)βˆˆβ„•π‘›0 satisfies 𝛾𝑗≀𝛽𝑗 for all 𝑗=1,…,𝑛, then we write 𝛾≀𝛽. Our first main result is about the strong derivatives of 𝐴(𝑧).

Theorem 1.1. Let Ξ© be a rotation invariant open set in ℂ𝑛 and πœ‡ a rotation invariant positive measure on Ξ© that satisfies (1.1) and has moments of every order. Moreover one assumes that, for any multi-index 𝛼 and any compact set 𝐾 of Ξ©, there exists 𝐺𝛼,𝐾∈𝐿2(Ξ©,πœ‡) such that ||π‘€π›Όπœ•π›Ό||𝐾(𝑧,𝑀)≀𝐺𝛼,𝐾(𝑀),βˆ€π‘§βˆˆπΎ.(1.12) Then for all 𝛽, 𝛾 multi-indices, the operators πœ•π›½πœ•π›Ύπ΄(𝑧) and πœ•π›½πœ•π›Ύπ΄(𝑧) are adjoint to each other; their rank is smaller or equal to the infimum of #{π›Ώβˆˆβ„•π‘›0,𝛿≀𝛾} and #{π›Ώβˆˆβ„•π‘›0,𝛿≀𝛽}.
Moreover, if at least 𝛽 or 𝛾 is different from 0, one has ξ‚ƒπœ•trπ›½πœ•π›Ύξ‚„π΄(𝑧)=0.(1.13)

Our second main result generalizes the estimates of Englis and Zhang for the strong derivatives of the Berezin transform on weighted Fock spaces.

Theorem 1.2. For a bounded linear operator 𝑇 on π’œ2(ℂ𝑛,πœ‡π‘š), the Berezin transform 𝑇 has derivatives of all orders. In addition to any multi-indices 𝛽, 𝛾, there exist positive constants 𝑐𝛽,𝛾 and 𝐴, depending on 𝛽, 𝛾, and 𝑛 only such that |||πœ•π›½πœ•π›Ύξ‚|||𝑇(𝑧)≀𝑐𝛽,𝛾|𝑧|((π‘š/2)βˆ’1)(|𝛽+𝛾|)for|𝑧|>𝐴.(1.14)

2. Preliminaries

We recall some properties of the Bergman kernel 𝐾, when πœ‡ is a positive rotation invariant measure on Ξ©. The kernel 𝐾 is given by𝐾(𝑧,πœ”)=π›½βˆˆβ„•π‘›01π‘π›½π‘§π›½πœ”π›½,(2.1) for 𝑧, πœ” in Ξ©, with the usual convention that, for 𝑧=(𝑧1,…,𝑧𝑛) and 𝛽=(𝛽1,…,𝛽𝑛), 𝑧𝛽 stands for 𝑧𝛽11⋯𝑧𝛽𝑛𝑛 and whereπ‘π›½ξ€œβˆΆ=Ξ©π‘₯𝛽π‘₯π›½π‘‘πœ‡(π‘₯).(2.2) By Lemma  2.1 in [4], we know that𝑐𝛽=(π‘›βˆ’1)!𝛽!π‘š|𝛽|ξ€·||𝛽||ξ€Έ!π‘›βˆ’1+,whereπ‘šπ‘˜=ξ€œΞ©|𝑧|2π‘˜π‘‘πœ‡(𝑧)forπ‘˜βˆˆβ„•0.(2.3) Thus we can write𝐾(𝑧,πœ”)=𝐹(βŸ¨π‘§,πœ”βŸ©),(2.4) where1𝐹(𝑑)∢=(ξ“π‘›βˆ’1)!π‘‘βˆˆβ„•0(π‘›βˆ’1+𝑑)!π‘šπ‘‘π‘‘π‘‘(2.5) is a holomorphic function of one complex variable.

For a bounded operator 𝑇 on π’œ2(Ξ©,πœ‡), the Berezin transform can be written in the form𝑇(𝑧)=tr(𝑇𝐴(𝑧)),forπ‘§βˆˆΞ©,(2.6) where 𝐴(𝑧) is the rank one projection 𝐴(𝑧)=βŸ¨β‹…,π‘˜π‘§βŸ©π‘˜π‘§.

Recall [3] that a mapping β„Ž from a domain in ℂ𝑛 into a Banach space possesses a strong holomorphic derivative πœ•β„Ž/πœ•π‘§1(𝑧) at a point 𝑧=(𝑧1,…,𝑧𝑛)βˆˆβ„‚π‘› iflim𝑑→0β€–β€–β€–β€–β„Žξ€·π‘§1+𝑑,𝑧2,…,π‘§π‘›ξ€Έξ€·π‘§βˆ’β„Ž1,…,π‘§π‘›ξ€Έπ‘‘βˆ’πœ•β„Žπœ•π‘§1𝑧1,…,𝑧𝑛‖‖‖‖=0;(2.7) similarly one can define the antiholomorphic derivative πœ•. Englis and Zhang showed that the mapping 𝑧↦𝐴(𝑧) has strong derivatives.

Lemma 2.1. Let Ξ© be a domain in ℂ𝑛, πœ‡ a measure on Ξ© that satisfies (1.1). Then the function π‘§β†¦π‘˜π‘§ has strong derivatives of all orders.

Lemma 2.2. Under the same hypothesis, the trace-class-operator-valued function π‘§βŸΌβŸ¨β‹…,π‘˜π‘§βŸ©π‘˜π‘§(2.8) from Ξ© into the space of trace-class operators on π’œ2(Ξ©,πœ‡) has strong derivatives of all orders.

The mapping 𝑧↦𝐴(𝑧) and its derivatives can be expressed in terms of the function 𝐹. It is easy tosee that, for 𝑧, 𝑒 in Ξ©,𝐴(𝑧)⋅𝑓(𝑒)=𝑓,𝐹(βŸ¨β‹…,π‘§βŸ©)𝐹(βŸ¨π‘’,π‘§βŸ©)𝐹(βŸ¨π‘§,π‘§βŸ©).(2.9)

3. Proof of Theorem 1.1

We write, for π‘“βˆˆπ’œ2(Ξ©,πœ‡) and 𝑧, 𝑒 in Ξ©,ξ‚Έξ€œπ΄(𝑧)⋅𝑓(𝑒)=Ω𝑓(πœ”)𝐹(βŸ¨π‘§,πœ”βŸ©)π‘‘πœ‡(πœ”)𝐹(βŸ¨π‘’,π‘§βŸ©)𝐹(βŸ¨π‘§,π‘§βŸ©).(3.1) Since it is possible to differentiate under the integral sign at any order with respect to 𝑧, for 𝛽 in ℕ𝑛0, we haveξ€·πœ•π›½ξ€Έξ€œπ΄(𝑧)⋅𝑓(𝑒)=Ω𝑓(πœ”)πœ•π›½ξ‚ΈπΉ(βŸ¨π‘§,πœ”βŸ©)𝐹(βŸ¨π‘§,π‘§βŸ©)π‘‘πœ‡(πœ”)𝐹(βŸ¨π‘’,π‘§βŸ©).(3.2) The Leibnitz rule leads toξ€·πœ•π›½ξ€Έξ“π΄(𝑧)⋅𝑓(𝑒)=πœˆβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ ξ€œΞ©π‘“(πœ”)πœ”πœˆπΉ(|𝜈|)(βŸ¨π‘§,πœ”βŸ©)π‘§π›½βˆ’πœˆξ‚€1𝐹(|π›½βˆ’πœˆ|)(βŸ¨π‘§,π‘§βŸ©)π‘‘πœ‡(πœ”)𝐹(βŸ¨π‘’,π‘§βŸ©).(3.3) Setting ̇𝐾𝑧(𝛽)(πœ”)=𝐹(βŸ¨π‘§,π‘§βŸ©)1/2βˆ‘πœˆβŠ‚π›½ξ€·π›½πœˆξ€Έπœ”πœˆπ‘§π›½βˆ’πœˆ(1/𝐹)(|π›½βˆ’πœˆ|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜈|)(βŸ¨πœ”,π‘§βŸ©), we getπœ•π›½ξ‚¬Μ‡πΎπ΄(𝑧)⋅𝑓=𝑓,𝑧(𝛽)ξ‚­π‘˜π‘§(3.4) and thenπœ•π›Ύπœ•π›½π΄ξ“(𝑧)⋅𝑓=𝛿≀𝛾𝑓,𝐿𝑧,𝛿𝑀𝑧,𝛿,(3.5) where 𝐿𝑧,𝛿(πœ”)=(πœ•π›Ώ/πœ•π‘§π›Ώ)(𝐹(βŸ¨π‘§,π‘§βŸ©)βˆ’1/2̇𝐾𝑧(𝛽)(πœ”)) and 𝑀𝑧,𝛿(𝑒)=π›Ύπ›Ώξ€Έπ‘’π›Ύβˆ’π›ΏπΉ(|π›Ύβˆ’π›Ώ|)(βŸ¨π‘’,π‘§βŸ©). Notice that the rank of the operator πœ•π›Ύπœ•π›½π΄(𝑧) is smaller than or equal to #{π›Ώβˆˆβ„•π‘›0,𝛿≀𝛾}.

Setting now π‘†βˆΆ=πœ•π›Ύπœ•π›½π΄(𝑧), πΏπ›ΏβˆΆ=𝐿𝑧,𝛿, and π‘€π›ΏβˆΆ=𝑀𝑧,𝛿 for a fixed element 𝑧 in Ξ©, we obtain, for 𝑓,𝑔 in π’œ2(Ξ©,πœ‡),ξ„”ξ“βŸ¨π‘†π‘“,π‘”βŸ©=π›Ώβ‰€π›ΎβŸ¨π‘“,πΏπ›ΏβŸ©π‘€π›Ώξ„•=,𝑔𝑓,π›Ώβ‰€π›ΎβŸ¨π‘”,π‘€π›ΏβŸ©πΏπ›Ώξ„•.(3.6) Consequently,π‘†βˆ—ξ“π‘”=π›Ώβ‰€π›ΎβŸ¨π‘”,π‘€π›ΏβŸ©πΏπ›Ώ,(3.7) that is,ξ€·π‘†βˆ—π‘”ξ€Έξ“(𝑒)=π›Ώβ‰€π›ΎβŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ€œΞ©π‘”(πœ”)πœ”π›Ύβˆ’π›ΏπΉ(|π›Ύβˆ’π›Ώ|)πœ•(βŸ¨π‘§,πœ”βŸ©)π‘‘πœ‡(πœ”)π›Ώπœ•π‘§π›Ώξƒ©1βˆšΜ‡πΎπΉ(βŸ¨π‘§,π‘§βŸ©)𝑧(𝛽)ξƒͺ=πœ•(𝑒)π›Ύπœ•π‘§π›Ύξƒ¬ξ€œΞ©ξƒ©1𝑔(πœ”)𝐹(βŸ¨π‘§,πœ”βŸ©)π‘‘πœ‡(πœ”)βˆšΜ‡πΎπΉ(βŸ¨π‘§,π‘§βŸ©)𝑧(𝛽)=πœ•(𝑒)ξƒͺξƒ­π›Ύπœ•π‘§π›Ύξ‚ƒβŸ¨π‘”,π‘˜π‘§βŸ©Μ‡πΎπ‘§(𝛽)ξ‚„=πœ•(𝑒)π›Ύπœ•π‘§π›Ύπœ•π›½πœ•π‘§π›½ξ€ΊβŸ¨π‘”,π‘˜π‘§βŸ©π‘˜π‘§ξ€»=ξ‚€πœ•(𝑒)π›Ύπœ•π›½π΄ξ‚(𝑧)⋅𝑔(𝑒).(3.8) It follows that the rank of the operator πœ•π›Ύπœ•π›½π΄(𝑧) is also smaller than or equal to #{π›Ώβˆˆβ„•π‘›0,𝛿≀𝛽}. Thusξ‚€πœ•π›Ύπœ•π›½ξ‚π΄(𝑧)βˆ—=πœ•π›Ύπœ•π›½π΄(𝑧),forπ‘§βˆˆΞ©.(3.9) Now let 𝛽 and 𝛾 be multi-indices. Due to Lemma 2.2 and the continuity of the linear form Tr(𝑋), we haveξ‚€πœ•Trπ›Ύπœ•π›½ξ‚π΄(𝑧)=πœ•π›Ύπœ•π›½Tr𝐴(𝑧)=πœ•π›Ύπœ•π›½1=𝛿𝛽0𝛿𝛾0.(3.10)

4. Proof of Theorem 1.2

When Ξ©=ℂ𝑛 and π‘‘πœ‡π‘š(𝑧)=π‘’βˆ’|𝑧|π‘šπ‘‘π‘£(𝑧), for brevity we set π’œ2(πœ‡π‘š) for π’œ2(ℂ𝑛,πœ‡π‘š). The Bergman kernel πΎπ‘š of π’œ2(πœ‡π‘š) can be expressed in terms of the Mittag-Leffler function (see [5]). Putting 𝛼=2/π‘š, we haveπΎπ‘š(𝑧,πœ”)=𝐹(βŸ¨π‘§,πœ”βŸ©),where𝐹(𝑑)=𝐢𝐸(π‘›βˆ’1)𝛼,π›Όπ‘š(𝑑),𝐢=,(π‘›βˆ’1)!(4.1)𝐸(π‘›βˆ’1)𝛼,𝛼 being the π‘›βˆ’1-th derivative of the Mittag-Leffler function 𝐸𝛼,𝛼, entire on β„‚ and defined by𝐸𝛼,𝛼(𝑑)∢=+βˆžξ“π‘‘=0𝑑𝑑Γ(𝛼𝑑+𝛼),forπ‘‘βˆˆβ„‚.(4.2) In what follows, we will use some asymptotic properties of this function (see [6]) near the positive real axis.

Lemma 4.1. Let 𝑝 a nonnegative integer. There exists a constant πœ–>0, such that, for any complex number 𝑧 with |arg𝑧|<πœ–, one has 𝐹(𝑝)(𝑧)=𝐹0(𝑝)(𝑧)+πœ–π‘(𝑧),(4.3) where 𝐹0(𝑧)=𝐢𝑒𝑍0(𝑧)𝑍0(𝑧)βˆ’π‘›π›Όπ‘ƒπ‘›ξ€·π‘0ξ€Έ(𝑧)(4.4) and 𝑍𝑠(𝑧)=|𝑧|1/𝛼exp[(𝑖/𝛼)(argz+2πœ‹π‘ )], for 𝑠 integer.
The polynomials (𝑃𝑝)π‘βˆˆβ„• are defined by (𝑑𝑝/𝑑𝑒𝑝)[𝑒𝑒1/𝛼]=(𝑒𝑒1/𝛼/𝑒𝑝)𝑃𝑝(𝑒1/𝛼), 𝑃𝑝 is of degree 𝑝, and when |𝑧|β†’+∞, πœ–π‘(π‘Ÿ)(𝑧)=π‘œ(𝐹0(π‘ž)(𝑍0)) for any nonnegative integers π‘Ÿ and π‘ž.

We also need asymptotic estimates for some auxiliaries functions.

Lemma 4.2. Let 𝑝 and 𝑑 be nonnegative integers. When 𝑑 is real and tends to +∞, ξ‚€1𝐹(𝑑)ξ‚΅1(𝑑)∼𝐹0ξ‚Ά(𝑑)𝐹(𝑑),(𝑝)𝐹(𝑑)𝐹(𝑑)∼0(𝑝)𝐹0ξƒͺ(𝑑)(𝑑).(4.5)

For 𝑇 a bounded linear operator on π’œ2(πœ‡π‘š), the Berezin transform is given by (see [3])𝑇(𝑧)=tr(𝑇𝐴(𝑧)).(4.6) Let 𝛽 and 𝛾 be some multi-indices. By differentiation, Lemma 2.2 givesπœ•π›½πœ•π›Ύξ‚ξ‚€π‘‡(𝑧)=trπ‘‡πœ•π›½πœ•π›Ύξ‚π΄(𝑧).(4.7) Due to the continuity of the bilinear form (𝑋,π‘Œ)↦tr(π‘‹π‘Œ), we have||||tr(π‘‹π‘Œ)β‰€β€–π‘‹β€–β€–π‘Œβ€–tr,(4.8) for all bounded operators 𝑋 and trace class operators π‘Œ. Therefore, we obtain|||πœ•π›½πœ•π›Ύξ‚|||β€–β€–πœ•π‘‡(𝑧)β‰€β€–π‘‡β€–π›½πœ•π›Ύβ€–β€–π΄(𝑧)tr.(4.9) Like in Section 3, we set π‘†βˆΆ=πœ•π›½πœ•π›Ύπ΄(𝑧). Next we shall estimate tr((π‘†π‘†βˆ—)1/2).

Fix 𝑓 in π’œ2(πœ‡π‘š) and 𝑧 in ℂ𝑛. We recall that𝑆𝑓=𝛿≀𝛾𝑓,𝐿𝑧,𝛿𝑀𝑧,𝛿,(4.10) where 𝐿𝑧,𝛿(πœ”)=(πœ•π›Ώ/πœ•π‘§π›Ώ)(𝐹(βŸ¨π‘§,π‘§βŸ©)βˆ’1/2̇𝐾𝑧(𝛽)(πœ”)) and 𝑀𝑧,𝛿(𝑒)=π›Ύπ›Ώξ€Έπ‘’π›Ύβˆ’π›ΏπΉ(|π›Ύβˆ’π›Ώ|)(βŸ¨π‘’,π‘§βŸ©). Thenπ‘†π‘†βˆ—ξ“π‘“=π›Ώβ‰€π›Ύξƒ¬ξ“π›Ώξ…žβ‰€π›Ύξ«π‘“,𝑀𝑧,𝛿′𝐿𝑧,𝛿′,𝐿𝑧,𝛿𝑀𝑧,𝛿.(4.11) To estimate the eigenvalues of the finite rank positive operator π‘†π‘†βˆ—, we compute its trace:ξ€·trπ‘†π‘†βˆ—ξ€Έ=ξ“π›Ώβ‰€π›Ύπ‘Žπ›Ώ,withπ‘Žπ›Ώ=ξ“π›Ώξ…žβ‰€π›Ύξ«π‘€π‘§,𝛿,𝑀𝑧,𝛿′𝐿𝑧,𝛿′,𝐿𝑧,𝛿.(4.12) With the operator being of finite rank, we observe that Tr(π‘†π‘†βˆ—)=Tr(π‘†βˆ—π‘†). We now compute and then estimate each diagonal term π‘Žπ›Ώ. Fix some multi-indices 𝛿 and 𝛿′ such that 𝛿≀𝛾 and 𝛿′≀𝛾. To simplify the notation we set 𝐿𝛿 and 𝑀𝛿 instead of 𝐿𝑧,𝛿 and 𝑀𝑧,𝛿. We obtain𝑀𝛿,𝑀𝛿′=𝐢2βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ ξ€œβ„‚π‘›πœ”π›Ύβˆ’π›ΏπΉ(|π›Ύβˆ’π›Ώ|)(βŸ¨πœ”,π‘§βŸ©)πœ”π›Ύβˆ’π›Ώβ€²πΉ(|π›Ύβˆ’π›Ώβ€²|)(βŸ¨π‘§,πœ”βŸ©)π‘‘πœ‡(πœ”)=𝐢2βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ 1πΆπœ•π›Ύβˆ’π›Ώπœ•π‘§π›Ύβˆ’π›Ώπœ•π›Ύβˆ’π›Ώβ€²πœ•π‘§π›Ύβˆ’π›Ώβ€²βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ πœ•πΉ(βŸ¨π‘§,π‘§βŸ©)=πΆπ›Ώβ€²π›Ύβˆ’π›Ώπœ•π‘§π›Ύβˆ’π›Ώξ‚ƒπ‘§π›Ύβˆ’π›Ώβ€²πΉ(|π›Ύβˆ’π›Ώβ€²|)ξ‚„βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ βŽ§βŽͺ⎨βŽͺβŽ©ξ“(βŸ¨π‘§,π‘§βŸ©)=πΆπ›Ώβ€²πœ…β‰€π›Ύβˆ’π›Ώ,πœ…β‰€π›Ύβˆ’π›Ώξ…žβŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ Γ—π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!π‘§π›Ύβˆ’π›Ώβ€²βˆ’πœ…π‘§π›Ύβˆ’π›Ώβˆ’πœ…πΉ(|π›Ύβˆ’π›Ώβ€²+π›Ύβˆ’π›Ώβˆ’πœ…|)⎫βŽͺ⎬βŽͺ⎭.(βŸ¨π‘§,π‘§βŸ©)(4.13) For the second term, sinceπΏπ›Ώπœ•(πœ”)=π›Ώπœ•π‘§π›ΏβŽ§βŽͺ⎨βŽͺβŽ©ξ“πœˆβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ πœ”π›½βˆ’πœˆπΉ(|π›½βˆ’πœˆ|)(βŸ¨πœ”,π‘§βŸ©)π‘§πœˆξ‚€1𝐹(|𝜈|)⎫βŽͺ⎬βŽͺ⎭=(βŸ¨π‘§,π‘§βŸ©)πœˆβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ πœ”π›½βˆ’πœˆπΉ(|π›½βˆ’πœˆ|)(πœ•βŸ¨πœ”,π‘§βŸ©)π›Ώπœ•π‘§π›Ώξ‚Έπ‘§πœˆξ‚€1𝐹(|𝜈|)(ξ‚Ή,βŸ¨π‘§,π‘§βŸ©)(4.14) we get𝐿𝛿′,𝐿𝛿=ξ“πœˆ,πœˆξ…žβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βŽžβŽŸβŽŸβŽ πœ•πœˆβ€²π›Ώξ…žπœ•π‘§π›Ώξ…žξ‚Έπ‘§πœˆβ€²ξ‚€1𝐹(|πœˆβ€²|)ξ‚Ήπœ•(βŸ¨π‘§,π‘§βŸ©)π›Ώπœ•π‘§π›ΏΓ—ξ‚Έπ‘§πœˆξ‚€1𝐹(|𝜈|)ξ‚Ήπœ•(βŸ¨π‘§,π‘§βŸ©)π›½βˆ’πœˆξ…žπœ•π‘§π›½βˆ’πœˆξ…žπœ•π›½βˆ’πœˆπœ•π‘§π›½βˆ’πœˆξ‚ΈπΉ(βŸ¨π‘§,π‘§βŸ©)𝐢,𝐿(4.15)𝛿′,𝐿𝛿=1πΆξ“πœˆ,πœˆξ…žβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βŽžβŽŸβŽŸβŽ πœ•πœˆβ€²π›Ώξ…žπœ•π‘§π›Ώξ…žξ‚Έπ‘§π›½βˆ’πœˆβ€²ξ‚€1𝐹(|π›½βˆ’πœˆβ€²|)ξ‚Ήπœ•(βŸ¨π‘§,π‘§βŸ©)π›Ώπœ•π‘§π›ΏΓ—ξ‚Έπ‘§π›½βˆ’πœˆξ‚€1𝐹(|π›½βˆ’πœˆ|)ξ‚Ήπœ•(βŸ¨π‘§,π‘§βŸ©)πœˆξ…žπœ•π‘§πœˆξ…žξ€Ίπ‘§πœˆπΉ(|𝜈|)ξ€».(βŸ¨π‘§,π‘§βŸ©)(4.16) By the Leibnitz rule, we see thatπœ•π›Ώπœ•π‘§π›Ώξ‚Έπ‘§π›½βˆ’πœˆξ‚€1𝐹(|π›½βˆ’πœˆ|)ξ‚Ή=(βŸ¨π‘§,π‘§βŸ©)πœ†β‰€π›Ώ,πœ†β‰€π›½βˆ’πœˆβŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ†βŽžβŽŸβŽŸβŽ π›½βˆ’πœˆπœ†!π‘§π›½βˆ’πœˆβˆ’πœ†π‘§π›Ώβˆ’πœ†ξ‚€1𝐹(|π›½βˆ’πœˆ+π›Ώβˆ’πœ†|)πœ•(βŸ¨π‘§,π‘§βŸ©),π›Ώξ…žπœ•π‘§π›Ώξ…žξ‚Έπ‘§π›½βˆ’πœˆβ€²ξ‚€1𝐹(|π›½βˆ’πœˆβ€²|)(ξ‚Ή=ξ“βŸ¨π‘§,π‘§βŸ©)πœ†ξ…žβ‰€π›Ώξ…ž,πœ†ξ…žβ‰€π›½βˆ’πœˆξ…žβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π›Ώβ€²πœ†β€²π›½βˆ’πœˆβ€²πœ†β€²ξ…ž!π‘§π›½βˆ’πœˆβ€²βˆ’πœ†β€²π‘§π›Ώβ€²βˆ’πœ†β€²Γ—ξ‚€1𝐹(|π›½βˆ’πœˆβ€²+π›Ώβ€²βˆ’πœ†β€²|)πœ•(βŸ¨π‘§,π‘§βŸ©),πœˆξ…žπœ•π‘§πœˆξ…žξ€Ίπ‘§πœˆπΉ(|𝜈|)(ξ€»=ξ“βŸ¨π‘§,π‘§βŸ©)πœŽβ‰€πœˆ,πœŽβ‰€πœˆβ€²βŽ›βŽœβŽœβŽπœˆξ…žπœŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœˆπœŽβŽžβŽŸβŽŸβŽ πœŽ!π‘§πœˆβˆ’πœŽπ‘§πœˆβ€²βˆ’πœŽπΉ(|𝜈+πœˆβ€²βˆ’πœŽ|)(βŸ¨π‘§,π‘§βŸ©).(4.17) Then 𝑀𝛿,𝑀𝛿′𝐿𝛿′,𝐿𝛿=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²πœˆβ‰€π›½,πœˆξ…žβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βŽžβŽŸβŽŸβŽ ξ“πœˆβ€²πœ†β‰€π›Ώ,π›½βˆ’πœˆπœ†ξ…žβ‰€π›Ώξ…ž,π›½βˆ’πœˆξ…žπœŽβ‰€πœˆ,πœˆξ…žπœ…β‰€π›Ύβˆ’π›Ώ,π›Ύβˆ’π›Ώξ…žβŽ›βŽœβŽœβŽπœŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœˆπœŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ Γ—βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ Γ—βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ πœˆβ€²πœŽ!π›Ώβ€²πœ†β€²π›½βˆ’πœˆβ€²πœ†β€²πœ†β€²!π›½βˆ’πœˆπœ†!π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!𝑧𝛽+π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœŽβˆ’πœ…π‘§π›½+π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœŽβˆ’πœ…Γ—πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜈+πœˆβ€²βˆ’πœŽ|)ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆβ€²+π›Ώβ€²βˆ’πœ†β€²|)ξ‚€1Γ—(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆ+π›Ώβˆ’πœ†|)(βŸ¨π‘§,π‘§βŸ©).(4.18) Thus π‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώξ…žβ‰€π›ΎβŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²πœˆβ‰€π›½πœˆξ…žβ‰€π›½βŽ›βŽœβŽœβŽπ›½πœˆβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βŽžβŽŸβŽŸβŽ ξ“πœˆβ€²πœ†β‰€π›Ώ,π›½βˆ’πœˆπœ†ξ…žβ‰€π›Ώξ…ž,π›½βˆ’πœˆξ…žπ›½βˆ’πœˆ,π›½βˆ’πœˆξ…žβ‰€πœŒπœ…β‰€π›Ύβˆ’π›Ώ,π›Ύβˆ’π›Ώξ…žβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœˆβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ πœˆβ€²π›½βˆ’πœŒπ›½βˆ’πœŒ(π›½βˆ’πœŒ)!π›Ώβ€²πœ†β€²π›½βˆ’πœˆβ€²πœ†β€²Γ—πœ†β€²!π›½βˆ’πœˆπœ†!π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!π‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒΓ—πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜈+πœˆβ€²βˆ’π›½+𝜌|)(ξ‚€1βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆβ€²+π›Ώβ€²βˆ’πœ†β€²|)(Γ—ξ‚€1βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆ+π›Ώβˆ’πœ†|)π‘Ž(βŸ¨π‘§,π‘§βŸ©),(4.19)𝛿=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²β‰€π›ΎβŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ ξ“πœ…ξ“πœ†,πœ†β€²π›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!ξ“πœˆ,πœˆβ€²πœŒβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ Γ—βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ (π›½βˆ’πœŒ)!πœŒβˆ’πœ†π›½βˆ’πœˆβˆ’πœ†πœŒβˆ’πœ†β€²π›½βˆ’πœˆβ€²βˆ’πœ†β€²π›Ώβ€²πœ†β€²π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!π‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒΓ—πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜈+πœˆβ€²βˆ’π›½+𝜌|)Γ—ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆβ€²+π›Ώβ€²βˆ’πœ†β€²|)ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆ+π›Ώβˆ’πœ†|)π‘Ž(βŸ¨π‘§,π‘§βŸ©),(4.20)𝛿=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²β‰€π›ΎβŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ ξ“πœ…βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώξ…žπœ…βŽžβŽŸβŽŸβŽ ξ“πœ…!𝜈,πœˆβ€²,πœ†,πœ†β€²,πœŒπ›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ Γ—βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώ(π›½βˆ’πœŒ)!ξ…žπœ†ξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœŒβˆ’πœ†π›½βˆ’πœˆβˆ’πœ†πœŒβˆ’πœ†ξ…žπ›½βˆ’πœˆξ…žβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ π‘§π›Ύβˆ’πœ†βˆ’πœ†β€²βˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†β€²βˆ’πœ…+πœŒΓ—πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜈+πœˆβ€²βˆ’π›½+𝜌|)Γ—ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆβ€²+π›Ώβ€²βˆ’πœ†β€²|)ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›½βˆ’πœˆ+π›Ώβˆ’πœ†|)(βŸ¨π‘§,π‘§βŸ©).(4.21) Setting 𝜎=π›½βˆ’πœˆβˆ’πœ† and πœŽβ€²=π›½βˆ’πœˆβ€²βˆ’πœ†β€², we haveπ‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²β‰€π›ΎβŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ ξ“πœ…βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώξ…žπœ…βŽžβŽŸβŽŸβŽ ξ“πœ…!πœ†β‰€π›Ώ,πœ†β€²β‰€π›Ώβ€²π›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώξ…žπœ†ξ…žβŽžβŽŸβŽŸβŽ Γ—ξ“πœ†β‰€πœŒ,πœ†β€²β‰€πœŒ(βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ ξ“πœŽπœŽβ‰€πœŒβˆ’πœ†,β€²β‰€πœŒβˆ’πœ†β€²βŽ›βŽœβŽœβŽπœŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœŒβˆ’πœ†πœŒβˆ’πœ†ξ…žπœŽξ…žβŽžβŽŸβŽŸβŽ Γ—πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝛽+πœŒβˆ’πœ†βˆ’πœ†β€²βˆ’πœŽβˆ’πœŽβ€²|)Γ—ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜎+𝛿|)ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|πœŽβ€²+𝛿′|)(βŸ¨π‘§,π‘§βŸ©)π‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+𝜌,π‘Ž(4.22)𝛿=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώξ…žβ‰€π›ΎβŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²πœ…βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ ξ“π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!πœ†β‰€π›Ώ,πœ†ξ…žβ‰€π›Ώξ…žπ›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ Γ—ξ“π›Ώβ€²πœ†β€²πœ†,πœ†β€²β‰€πœŒβ‰€π›½βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽ(π›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ π‘§π›Ύβˆ’πœ†βˆ’πœ†β€²βˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†β€²βˆ’πœ…+πœŒπ‘†πœŒ(βŸ¨π‘§,π‘§βŸ©),(4.23) whereπ‘†πœŒ=ξ“πœŽβ‰€πœŒβˆ’πœ†πœŽξ…žβ‰€πœŒβˆ’πœ†ξ…žβŽ›βŽœβŽœβŽπœŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πΉπœŒβˆ’πœ†πœŒβˆ’πœ†β€²πœŽβ€²(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)𝐹(|𝛽+πœŒβˆ’πœ†βˆ’πœ†β€²βˆ’πœŽβˆ’πœŽβ€²|)ξ‚€1𝐹(|𝜎+𝛿|)ξ‚€1𝐹(|πœŽβ€²+𝛿′|).(4.24) Setting Μ†πœ=(𝜏1,…,πœπ‘›βˆ’1) for a multiindex 𝜏=(𝜏1,…,πœπ‘›), we see thatπ‘†πœŒ=𝐹(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)ξ“Μ†πœ†Μ†Μ†Μ†πœŽβ‰€Μ†πœŒβˆ’πœŽξ…žβ‰€Μ†πœŒβˆ’πœ†ξ…žβŽ›βŽœβŽœβŽΜ†πœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽΜ†Μ†βŽžβŽŸβŽŸβŽ ξ“Μ†πœŒβˆ’Μ†πœŽΜ†πœŒβˆ’πœ†β€²πœŽβ€²πœŽβ€²π‘›β‰€πœŒπ‘›βˆ’πœ†β€²π‘›βŽ›βŽœβŽœβŽπœŒπ‘›βˆ’πœ†ξ…žπ‘›πœŽξ…žπ‘›βŽžβŽŸβŽŸβŽ ξ‚€1𝐹(|πœŽβ€²+𝛿′|)Γ—ξ“πœŽπ‘›β‰€πœŒπ‘›βˆ’πœ†π‘›βŽ›βŽœβŽœβŽπœŒπ‘›βˆ’πœ†π‘›πœŽπ‘›βŽžβŽŸβŽŸβŽ ξ‚΅ξ‚€1𝐹(|𝛿|+𝜎1+β‹―+πœŽπ‘›βˆ’1)ξ‚Ά(πœŽπ‘›)ξ‚€(𝐹)(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²Μ†|+|Μ†πœŒβˆ’πœ†βˆ’Μ†πœŽ|)(πœŒπ‘›βˆ’πœ†π‘›βˆ’πœŽπ‘›).(4.25) Using the Leibnitz ruleπ‘†πœŒ=𝐹(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)ξ“Μ†πœ†Μ†Μ†Μ†πœŽβ‰€Μ†πœŒβˆ’πœŽξ…žβ‰€Μ†πœŒβˆ’πœ†ξ…žβŽ›βŽœβŽœβŽΜ†πœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽΜ†Μ†βŽžβŽŸβŽŸβŽ ξ“Μ†πœŒβˆ’Μ†πœŽΜ†πœŒβˆ’πœ†β€²πœŽβ€²πœŽβ€²π‘›β‰€πœŒπ‘›βˆ’πœ†β€²π‘›βŽ›βŽœβŽœβŽπœŒπ‘›βˆ’πœ†ξ…žπ‘›πœŽξ…žπ‘›βŽžβŽŸβŽŸβŽ ξ‚€1𝐹(|πœŽβ€²+𝛿′|)Γ—ξ‚Έξ‚€1𝐹(|𝛿|+𝜎1+β‹―+πœŽπ‘›βˆ’1)𝐹(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²Μ†|+|Μ†πœŒβˆ’πœ†βˆ’Μ†πœŽ|)ξ‚Ή(πœŒπ‘›βˆ’πœ†π‘›).(4.26) An induction process shows thatπ‘†πœŒ=𝐹(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)ξ“πœŽξ…žβ‰€πœŒβˆ’πœ†ξ…žβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ ξ‚€1πœŒβˆ’πœ†β€²πœŽβ€²πΉξ‚(|πœŽβ€²+𝛿′|)𝐹(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|).(4.27) Thereforeπ‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ώξ…žβ‰€π›ΎβŽ›βŽœβŽœβŽπ›ΎβŽžβŽŸβŽŸβŽ ξ“π›Ώβ€²πœ…βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπœ…βŽžβŽŸβŽŸβŽ ξ“π›Ύβˆ’π›Ώπ›Ύβˆ’π›Ώβ€²πœ…!πœ†β‰€π›Ώ,πœ†ξ…žβ‰€π›Ώξ…žπ›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ Γ—ξ“π›Ώβ€²πœ†β€²πœ†,πœ†β€²β‰€πœŒβ‰€π›½(βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ πΉ(|π›Ύβˆ’π›Ώ+π›Ύβˆ’π›Ώβ€²βˆ’πœ…|)(βŸ¨π‘§,π‘§βŸ©)π‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒπ‘§π›Ύβˆ’πœ†βˆ’πœ†ξ…žβˆ’πœ…+πœŒΓ—ξ“πœŽβ€²β‰€πœŒβˆ’πœ†β€²βŽ›βŽœβŽœβŽπœŒβˆ’πœ†ξ…žπœŽξ…žβŽžβŽŸβŽŸβŽ ξ‚€1𝐹(|πœŽβ€²+𝛿′|)𝐹(βŸ¨π‘§,π‘§βŸ©)(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|)(βŸ¨π‘§,π‘§βŸ©).(4.28) Now let 𝜏=π›Ύβˆ’πœ†β€²βˆ’πœ…. It follows thatπ‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“πœŒβ‰€π›½ξ“πœβ‰€π›Ύξ“πœ†,πœ†β€²β‰€πœŒξ“πœ†β€²β‰€π›Ώβ€²β‰€πœ+πœ†β€²βŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύβˆ’π›Ώπ›Ύβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβˆ’πœπ›Ύβˆ’π›Ώξ…žπ›Ύβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ ξ€·βˆ’πœπ›Ύβˆ’πœ†ξ…žξ€Έ!βˆ’πœπ›½!(π›½βˆ’πœ†)!𝛽!ξ€·π›½βˆ’πœ†ξ…žξ€Έ!Γ—βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώξ…žπœ†ξ…žβŽžβŽŸβŽŸβŽ (βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†ξ…žπœŒβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ πΉ(|π›Ύβˆ’π›Ώβˆ’π›Ώβ€²+πœ†β€²+𝜏|)(ξ“βŸ¨π‘§,π‘§βŸ©)πœŽβ€²β‰€πœŒβˆ’πœ†β€²βŽ›βŽœβŽœβŽπœŒβˆ’πœ†ξ…žπœŽξ…žβŽžβŽŸβŽŸβŽ Γ—ξ‚€1𝐹(|πœŽβ€²+𝛿′|)𝐹(βŸ¨π‘§,π‘§βŸ©)(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|)(βŸ¨π‘§,π‘§βŸ©)π‘§πœπ‘§πœπ‘§πœŒβˆ’πœ†π‘§πœŒβˆ’πœ†.(4.29) On the other handβŽ›βŽœβŽœβŽπ›Ύπ›Ώξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ύβˆ’π›Ώξ…žπ›Ύβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ ξ€·βˆ’πœπ›Ύβˆ’πœ†ξ…žξ€Έ!βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ =βˆ’πœπ›Ώβ€²πœ†β€²π›Ύ!ξ€·βˆ’π›Ώξ…ž+πœ†ξ…žξ€Έ+𝜏!πœ†ξ…ž!ξ€·π›Ώξ…žβˆ’πœ†ξ…žξ€Έ!=𝛾!βŽ›βŽœβŽœβŽπœβŽžβŽŸβŽŸβŽ .𝜏!πœ†β€²!π›Ώβ€²βˆ’πœ†β€²(4.30) Thus, if we set 𝜎=π›Ώβ€²βˆ’πœ†β€², thenπ‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“πœŒβ‰€π›½ξ“πœβ‰€π›Ύξ“πœ†,πœ†β€²πœŽβ‰€πœŒβ€²β‰€πœŒβˆ’πœ†β€²ξ“πœŽβ‰€πœβŽ›βŽœβŽœβŽπœŒβˆ’πœ†ξ…žπœŽξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ (π›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†β€²πœŒβˆ’πœ†β€²π›Ύ!π‘§πœ+πœŒβˆ’πœ†π‘§πœ+πœŒβˆ’πœ†Γ—βŽ›βŽœβŽœβŽπ›Ύβˆ’π›Ώπ›Ύβˆ’πœ†ξ…žβŽžβŽŸβŽŸβŽ 1βˆ’πœπœ!πœ†ξ…ž!βŽ›βŽœβŽœβŽπœπœŽβŽžβŽŸβŽŸβŽ πΉ(|π›Ύβˆ’π›Ώ+πœβˆ’πœŽ|)Γ—ξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|𝜎+πœŽβ€²+πœ†β€²|)(ξ‚ΈπΉβŸ¨π‘§,π‘§βŸ©)(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|)(βŸ¨π‘§,π‘§βŸ©).(4.31) Using the identityξ“πœŽβ‰€πœβŽ›βŽœβŽœβŽπœπœŽβŽžβŽŸβŽŸβŽ ξ€·πΉ(|π›Ύβˆ’π›Ώ|)ξ€Έ(|πœβˆ’πœŽ|)ξ‚΅ξ‚€1𝐹(|πœŽβ€²+πœ†β€²|)ξ‚Ά(|𝜎|)=ξ‚Έξ‚€1𝐹(|π›Ύβˆ’π›Ώ|)𝐹(|πœŽβ€²+πœ†β€²|)ξ‚Ή(|𝜏|),(4.32) we see thatπ‘Žπ›Ώ=βŽ›βŽœβŽœβŽπ›Ύπ›ΏβŽžβŽŸβŽŸβŽ ξ“π›Ύ!πœŒβ‰€π›½πœβ‰€π›Ύξ“πœ†,πœ†β€²πœŽβ‰€πœŒβ€²β‰€πœŒβˆ’πœ†β€²βŽ›βŽœβŽœβŽπœŒβˆ’πœ†ξ…žπœŽξ…žβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Ώπœ†βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ 1(π›½βˆ’πœŒ)!π›½βˆ’πœ†πœŒβˆ’πœ†π›½βˆ’πœ†β€²πœŒβˆ’πœ†β€²π›Ύβˆ’π›Ώπ›Ύβˆ’πœ†β€²βˆ’πœπœ!πœ†ξ…ž!π‘§πœ+πœŒβˆ’πœ†π‘§πœ+πœŒβˆ’πœ†Γ—ξ‚ΈπΉ(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|)ξ‚Έξ‚€1(βŸ¨π‘§,π‘§βŸ©)𝐹(|π›Ύβˆ’π›Ώ|)𝐹(|πœŽβ€²+πœ†β€²|)ξ‚Ή(|𝜏|)(βŸ¨π‘§,π‘§βŸ©).(4.33) Notice that, for 𝑝 and π‘ž integers, we have, when 𝑑 is real and 𝑑→+∞,𝐹(𝑝)(𝑑)𝐹(𝑑)βˆΌπ‘‘((1/𝛼)βˆ’1)𝑝,ξ‚€1𝐹(π‘ž)βˆΌπ‘‘(𝑑)((1/𝛼)βˆ’1)𝑝.𝐹(𝑑)(4.34) Hence by Lemmas 4.1 and 4.2 we have the following estimates:||||𝐹(|π›½βˆ’πœ†β€²βˆ’πœŽβ€²|)ξ‚€1𝐹(|𝛿|)ξ‚Ή(|πœŒβˆ’πœ†|)||||(𝑑)≀𝐢𝑑((1/𝛼)βˆ’1)(|𝛽+π›Ώβˆ’πœ†β€²βˆ’πœŽβ€²|)βˆ’|πœŒβˆ’πœ†|,||||ξ‚Έξ‚€1𝐹(|π›Ύβˆ’π›Ώ|)𝐹(|πœŽβ€²+πœ†β€²|)ξ‚Ή(|𝜏|)||||(𝑑)≀𝐢𝑑((1/𝛼)βˆ’1)(|π›Ύβˆ’π›Ώ+πœŽβ€²+πœ†β€²|)βˆ’|𝜏|(4.35) for some constant 𝐢 when the real 𝑑 tends to +∞. Thus for any multiindex 𝛿≀𝛾, there exists a constant 𝐢′ such that π‘Žπ›Ώβ‰€πΆξ…ž|𝑧|2((1/𝛼)βˆ’1)(|𝛽+𝛾|). Then taking the sum over 𝛿 we getξ€·trπ‘†π‘†βˆ—ξ€Έβ‰€πΆξ…žξ…ž|𝑧|2((1/𝛼)βˆ’1)(|𝛽+𝛾|).(4.36) Since Tr(π‘†π‘†βˆ—)=Tr(π‘†βˆ—π‘†) and the operator π‘†βˆ—π‘† is positive, its eigenvalues are bounded above by πΆξ…žξ…ž|𝑧|2((1/𝛼)βˆ’1)(|𝛽+𝛾|). Thus the eigenvalues of (π‘†π‘†βˆ—)1/2 are bounded above by πΆξ…žξ…žξ…ž|𝑧|((1/𝛼)βˆ’1)(|𝛽+𝛾|). Since (π‘†βˆ—π‘†)1/2 is of finite rank, the proof is complete.

Acknowledgments

The author wishes to thank El Hassan Youssfi and Miroslav Englis for useful discussions.

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